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- W3159456048 abstract "Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial <math xmlns=http://www.w3.org/1998/Math/MathML id=M1> <mi>p</mi> </math> with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to <math xmlns=http://www.w3.org/1998/Math/MathML id=M2> <mi>p</mi> </math> if we restrict the coefficients to be real. Let <math xmlns=http://www.w3.org/1998/Math/MathML id=M3> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M4> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> be the vector space of all polynomials of degree <math xmlns=http://www.w3.org/1998/Math/MathML id=M5> <mi>n</mi> </math> or less with real coefficients. In this article, we give explicit forms of polynomials in <math xmlns=http://www.w3.org/1998/Math/MathML id=M6> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on <math xmlns=http://www.w3.org/1998/Math/MathML id=M7> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> which preserve real roots of polynomials in a certain subset of <math xmlns=http://www.w3.org/1998/Math/MathML id=M8> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> ." @default.
- W3159456048 created "2021-05-10" @default.
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- W3159456048 date "2021-04-30" @default.
- W3159456048 modified "2023-10-18" @default.
- W3159456048 title "Real Root Polynomials and Real Root Preserving Transformations" @default.
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- W3159456048 doi "https://doi.org/10.1155/2021/5585480" @default.
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